In Memory of Wenbo V Li s Contributions

Transcription

1 In Memory of Wenbo V Li s Contributions Qi-Man Shao The Chinese University of Hong Kong qmshao@cuhk.edu.hk The research is partially supported by Hong Kong RGC GRF

2 Outline Lower tail probabilities Capture time of Brownian pursuits Random polynomials Normal comparison inequalities Wenbo s weak correlation inequality

3 1. Lower tail probabilities Let {X t, t T} be a real valued Gaussian process indexed by T with E X t = 0. Lower tail probability refers to ) P supx t X t0 ) x t T as x 0, t 0 T or Examples: ) P sup X t x t T a) Capture time of Brownian pursuits b) Random polynomials as T

4 Li and Shao 2004): Established a general result for the lower tail probability of non-stationary Gaussian process ) P supx t X t0 ) x t T as x 0,

5 Special cases: Let {Xt), t [0, 1] d } be a centered Gaussian process with X0) = 0 and stationary increments, that is t, s [0, 1] d, E X t X s ) 2 = σ 2 t s ). If there are 0 < α β < 1 such that σh)/h α, σh)/h β Then there exist 0 < c 1 c 2 < such that for 0 < x < 1/2 c 2 ln 1 ) x ln P sup Xt) σx) c 1 ln 1 t [0,1] d x.

6 Special cases: Let {Xt), t [0, 1] d } be a centered Gaussian process with X0) = 0 and stationary increments, that is t, s [0, 1] d, E X t X s ) 2 = σ 2 t s ). If there are 0 < α β < 1 such that σh)/h α, σh)/h β Then there exist 0 < c 1 c 2 < such that for 0 < x < 1/2 c 2 ln 1 ) x ln P sup Xt) σx) c 1 ln 1 t [0,1] d x. In particular, for the fractional Levy s Brownian motion L α t) of order α, i.e. L α 0) = 0 and E L α t) L α s)) 2 = t s 2α, ) ln P sup L α t) x ln 1 t [0,1] d x.

7 Let {Xt), t [0, 1] d } be a centered Gaussian process with X0) = 0 and E X t X s ) = d i=1 If there are 0 < α β < 1 such that 1 2 σ2 t i ) + σ 2 s i ) σ 2 t i s i )). σh)/h α, σh)/h β Then ) ln P sup Xt) σ d x) ln d 1 t [0,1] d x.

8 Let {Xt), t [0, 1] d } be a centered Gaussian process with X0) = 0 and E X t X s ) = d i=1 If there are 0 < α β < 1 such that 1 2 σ2 t i ) + σ 2 s i ) σ 2 t i s i )). σh)/h α, σh)/h β Then ) ln P sup Xt) σ d x) t [0,1] d ln d 1 x. In particular, for d-dimensional fractional Brownian sheet B α t) i.e., σh) = h α ) ) ln P sup B α t) x ln d 1 t [0,1] d x

9 Li and Shao 2004): Obtained a connection for the lower tail probabilities between a non-stationary Gaussian process and a dual-stationarized Gaussian processes.

10 Explicit bounds of lower tail probabilities Li and Xiao 2013)) Let Bt) be the Brownian motion. For 0 < θ < 1 and P sup Bt) θb1)) x) 0 t 1 1 3θ 2 1 θ) 2 2π x3, P sup Bt) θtb1)) x) 1 θ) 2/π x 0 t 1 ln P sup Bt) 0 t 1 1 where c is a specified constant. 0 Bs)ds x) x 2 c

11 2. Capture time of Brownian pursuits Let B 0, B 1,, B n be independent standard Brownian motions. Define { } τ n = inf t > 0 : max B kt) = B 0 t) k n When is E τ n ) finite?

12 2. Capture time of Brownian pursuits Let B 0, B 1,, B n be independent standard Brownian motions. Define { } τ n = inf t > 0 : max B kt) = B 0 t) k n When is E τ n ) finite? Note that for any a > 0, by Brownian scaling, Pτ n > t) = P max = P 1 k n max 1 k n ) sup B k s) B 0 s)) < 1 0 s t sup B k s) B 0 s)) < t 1/2). 0 s 1 Thus, the estimate is reduced to a lower tail probability problem.

13 DeBlassie 1987): P{τ n > t} ct γn as t. Bramson and Griffeath 1991): E τ 3 = Li and Shao 2001): E τ 5 <. Ratzkin and Treibergs 2009): E τ 4 <.

14 What is the asymptotic behavior of γ n? Kesten 1992): 0 < lim inf γ n/ ln n lim sup γ n / ln n 1/4 n Conjecture: lim n γ n / ln n exists. n Li and Shao 2002): lim γ n/ ln n = 1/4 n

15 Li and Shao 2002) also consider the capture time of the fractional Brownian motion pursuit. Let {B k,α t); t 0}k = 0, 1, 2,..., n) be independent fractional Brownian motions of order α 0, 1). Put { } τ n := τ n,α = inf t > 0 : max B k,αt) = B 0,α t) k n Let and X k,α t) = e tα B k,α e t ), k = 0, 1,, n 1 γ n,α := lim T T ln P sup 0 t T ) max X k,αt) 0 1 k n

16 Li and Shao 2002): 1 γ n,α lim inf d α n ln n lim sup n γ n,α ln n <, where d α = 2 0 e2αx + e 2αx e x e x ) 2α )dx.

17 Li and Shao 2002): 1 γ n,α lim inf d α n ln n lim sup n γ n,α ln n <, where d α = 2 0 e2αx + e 2αx e x e x ) 2α )dx. Conjecture: γ n,α lim n ln n = 1. d α

18 3. The probability that a random polynomial has no real root Dembo, Poonen, Shao and Zeitouni 2002) n P Z i x i < 0 x R 1) = n b+o1) i=0 where n is even, Z i are i.i.d. N0, 1), and 1 b = 4 lim T T ln P sup 0 t T ) Xt) 0 where Xt) is a centered stationary Gaussian process with E Xs)Xt) = 2e t s /2 1 + e t s

19 Dembo, Poonen, Shao and Zeitouni 2002): 0.4 < b < Simulation: b = 0.76 ± 0.03

20 Dembo, Poonen, Shao and Zeitouni 2002): 0.4 < b < Simulation: b = 0.76 ± 0.03 Li and Shao 2002): 0.5 < b < 1

21 Let {Bt), t 0} be the Brownian motion and put B 0 t) = Bt), Li and Shao 2003+) and B m t) = t 0 B m 1 s)ds P sup B m t) x) = x rm+o1) 0 t 1 b 2r m 2m + 1), 2r m 2m + 1) b

22 Let {Bt), t 0} be the Brownian motion and put B 0 t) = Bt), Li and Shao 2003+) and B m t) = t 0 B m 1 s)ds P sup B m t) x) = x rm+o1) 0 t 1 b 2r m 2m + 1), 2r m 2m + 1) b Open questions: What is the value of b? If {Xt), t 0} is a differentiable stationary Gaussian process with positive correlation, what is the limit 1 ) lim T T ln P sup Xt) 0? 0 t T

23 4. Comparison inequalities Let n 2, and let ξ j, 1 j n) be standard normal random variables with correlation matrix R 1 = r 1 ij ), let η j, 1 j n) be standard normal random variables with correlation matrix R 0 = r 0 ij ). Set ρ ij = max r 1 ij, r0 ij ). Slepian s Lemma: If r 1 ij r0 ij, then n ) n ) P {ξ j u j } P {η j u j } j=1 j=1

24 Normal comparison inequality: Berman 1964,1971), Cramér and Leadbetter 1967)): n ) n ) P {ξ j u j } P {η j u j } j=1 1 2π 1 i<j n j=1 r 1 ij r 0 ij) + 1 ρ 2 ij) 1/2 exp u2 i + u 2 ) j 21 + ρ ij )

25 Normal comparison inequality: Berman 1964,1971), Cramér and Leadbetter 1967)): n ) n ) P {ξ j u j } P {η j u j } j=1 1 2π 1 i<j n Li and Shao 2002): j=1 r 1 ij r 0 ij) + 1 ρ 2 ij) 1/2 exp n ) n ) P {ξ j u j } P {η j u j } j= i<j n j=1 r 1 ij r 0 ij) + exp u2 i + u 2 ) j 21 + ρ ij ) u2 i + u 2 ) j 21 + ρ ij )

26 Li and Shao 2002): If r 1 ij r 0 ij 0 for all 1 i, j n Then n ) P {η j u j } j=1 n ) n ) P {ξ j u j } P {η j u j } j=1 { exp 1 i<j n ln j=1 for any u i 0, i = 1, 2,, n satisfying π 2 arcsinr 0 ij )) π 2 arcsinrij 1) exp u2 i + u 2 j ) )} 21 + rij 1) r l ki rl ijr l kj )u i + r l kj rl ijr l ki )u j 0 for l = 0, 1 and for all 1 i, j, k n.

27 In particular, for u 0 n ) P {η j u} j=1 n ) n ) P {ξ j u} P {η j u} j=1 { exp 1 i<j n ln j=1 π 2 arcsinr 0 ij )) π 2 arcsinrij 1) exp u2 )} 1 + rij 1

28 5. Wenbo s weak correlation inequality Let X and Y be centered jointly Gaussian vectors in a separable Banach space E. Gaussian correlation conjecture: For any symmetric convex sets A and B in E PX A, Y B) PX A)PY B)

29 5. Wenbo s weak correlation inequality Let X and Y be centered jointly Gaussian vectors in a separable Banach space E. Gaussian correlation conjecture: For any symmetric convex sets A and B in E PX A, Y B) PX A)PY B) Li 1999): For any 0 < λ < 1 PX A, Y B) PX λa)py 1 λ 2 ) 1/2 B)

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